The misalignment fault is one of the most common faults of the aero-engine rotor system. In actual production, support misalignment fault of rotor system is caused by the manufacturing error, installation error and thermal expansion of the bearing base, etc. Misalignment fault will lead to excessive stress of rotor shaft and instability of system motion. It can also cause other faults in the rotor system. The research on the influence of misalignment faults to dual-rotor system vibration response characteristics occupies an important role in the diagnosis and identification of aero-engine misalignment faults.

In recent years, the research on the dynamic characteristics of the rotor system with misalignment has been conducted by relevant scholars. Nanfei Wang and Dongxiang Jiang(2018), Han Qingkai et al (2016), Zhang Hongxian et al (2019), Liu Zhansheng et al (2007) and A.A.Pachpor et al (2016) summarized the misalignment of the rotor system. The rotor misalignment fault is mainly divided into coupling misalignment and support misalignment. As early as 1976, Gibbon C B. (1976) deduced and obtained the calculation formula of the force and torque generated at the connection point between the coupling and the rotating shaft. Sekhar and Prabhu (1995) derived the calculation formula of angular misalignment force based on Gibbons formula. Lee YS and Lee C-W (1999) derived the formula of excitation force decomposition when the coupling on the rotor system is the misalignment, and the vibration response of the rotor system was analyzed. Al-Hussain and Redmond (2002) derived the kinematics differential equation of the rotor system with parallel misalignment of rigid coupling used the basic kinematic principle, based on the Lagrange equation. Chen Guo and Li Xingyan (2009) studied the vibration characteristics of rotor system under sleeve coupling misalignment faults. Some researchers (Arumugam P et al 1997) studied the misalignment excitation when the bearing of the rotor is misalignment, and further analyzed the influence of the misalignment of the bearing on the dynamics of the rotor system. Feng Guoquan et al.(2012)took the double rotor system of an aero-engine as the research object and considered the misalignment of outer rotor bearing, established the dynamic model of the dual-rotor system by using the finite element method and studied the vibration response characteristics of the rotor under different rotating speeds and misalignment through numerical simulation. Dong Xiao (2010) deduces the dynamic models of misalignment of supporting bearings and intermediate bearing in the dual-rotor system by means of the finite element method. Xu Meipeng et al. (2019) equated the eccentricity of the high-pressure rotor support with the additional misalignment moment at the support of the high-pressure rotor and deduced the equation of the dual-rotor system considering the eccentricity of the high-pressure rotor support by using the Lagrange equation method.

Research on rotor misalignment mainly focuses on coupling misalignment in the rotor system (Sun C et. al 2004, Gao Hongtao et. al 2003, Zhao G et. al 2011, Li Ming et. al 2002, Liu ZS et. al 2007). The team (Yulin Jin et. al 2021, Kuan Lu et. al 2021) established a simplified model of a dual rotor system with four supports and four disks. The proper orthogonal decomposition method is used to reduce the dimension of the dual-rotor system for the first time. With the help of the finite element method, the nonlinear response caused by the misalignment of the coupling between the rotor and the motor was studied and verified by the test method. Research on the misalignment of the support is basically aimed at misalignment of bearing, but misalignment of bearing is only a part of misalignment of the support, and lack of research on a misalignment of rolling bearing. The research on a misalignment of support has theoretical and practical engineering needs.

Although many scholars have done a lot of researches on the vibration of rotor system with misalignment fault, in the current research, the modeling of coupling misalignment and bearing misalignment is relatively mature, and the impact on rotor system vibration is mainly concentrated on a single rotor. For the dual-rotor system, the vibration coupling problem of the dual-rotor system considering the support misalignment fault and the influence of the fault on the coupling is seldom involved. In this paper, a typical aero-engine model tester is taken as the research object, considering the load torque force of the low-pressure caused by the misalignment of the rear support of the low-pressure turbine, the excitation mechanism of support misalignment is analyzed. The motion differential equation of the dual-rotor system with misalignment faults is established, and the effects of misalignment, unbalanced mass and load torque on the vibration response of the dual-rotor are studied. Finally, test verification is carried out by the designed dual-rotor test rig.

### The Dynamic Model For A Dual-rotor System

With the increase of the thrust-weight ratio of aero-engine, the structure of the rotor system changes greatly, and the high speed of aero-engine makes the vibration problem of aero-engine more prominent. The coupling characteristics of aero-engine rotor dynamics and the transfer law of rotor misalignment vibration between the high-pressure rotor and the low-pressure rotor are still unclear, which needs to be studied urgently. In this paper, an advanced aero-engine rotor in reference(Ma P et. al 2021)was simplified as a rigid rotor with elastic supports, as shown in Fig. 1. The inner rotor is also called low-pressure rotor, and the outer rotor is also called high-pressure rotor. The low-pressure rotor consists of one fan (referred to as LPC) and one turbine (referred to as LPT), supported by bearings of 1, 2, 4, and 5. The high-pressure rotor is composed of one compressor disc (referred to as HPC) and a high-pressure turbine (referred to as HPT), which is supported by bearings of 3 and 4.

The parameters of the dual-rotor system are shown in Table 1–4, among which the parameters of four equivalent discs are shown in Table 1, the equivalent parameters of the rotating shaft are shown in Table 2, the distance parameters are shown in Table 3. By adjusting stiffness of supporting, the simplified model has dynamic similarity with the dual-rotor system of the prototype. The designed stiffness parameters are shown in Table 4.

Table 1

Disc parameters of the dynamic model

Disc | Thickness | External diameter | Internal diameter | Mass | Moment of inertia(kg·m^2) |

LPC | 18 mm | 330 mm | 35 mm | 14.0 kg | *J**p*=0.164 *J**d*=0.101 |

LPT | 10 mm | 316 mm | 48 mm | 11.3 kg | *J**p*=0.059 *J**d*=0.060 |

HPC | 35 mm | 305 mm | 73 mm | 20 kg | *J**p*=0.230 *J**d*=0.120 |

HPT | 14 mm | 304 mm | 62 mm | 10 kg | *J**p*=0.102 *J**d*=0.050 |

Table 2

Shaft parameters of the dual-rotor system

Shaft | External diameter | Internal diameter | Length |

Fan-rotor | 30 mm | 0 mm | 381 mm |

Low-pressure turbine shaft | 30 mm | 0 mm | 923 mm |

High-pressure shaft | 65 mm | 40 mm | 680 mm |

Table 3

Distance parameters of of dual rotor system

LAB | LAC | LAD | LAE | LAF | LAG | LAH | LAI |

127mm | 372 mm | 488 mm | 610 mm | 1000 mm | 1069 mm | 1109 mm | 1186 mm |

Table 4

Equivalent stiffness of five supports

*k**b1* | *k**b2* | *k**b3* | *k**b4* | *k**b5* |

0.9×107 N/m | 2.5×108 N/m | 0.3×107 N/m | 2.5×108 N/m | 0.3×107 N/m |

### Motion Equations Of The Dual-rotor System

The motion equations of the dual-rotor system as follows.

$$\frac{d}{{dt}}\frac{{\partial {\mathbf{T}}}}{{\partial {{{\mathbf{\dot {q}}}}_{\mathbf{j}}}}} - \frac{{\partial {\mathbf{T}}}}{{\partial {{\mathbf{q}}_{\mathbf{j}}}}}+\frac{{\partial {\mathbf{U}}}}{{\partial {{\mathbf{q}}_{\mathbf{j}}}}}={{\mathbf{Q}}_{\mathbf{j}}}(t),{\text{ }}j=1,2,3 \cdots$$

1

Where, the kinetic energy of the system can be recorded as follows.

$$T={T_{t1}}+{T_{r1}}+{T_{t2}}+{T_{r2}}+{T_{t3}}+{T_{r3}}+{T_{t4}}+{T_{r4}}$$

2

\({T_{ti}}\) and \({T_{ri}}\) are the translational kinetic energy and rotational kinetic energy of disc respectively.

The total potential energy of the system can be recorded as follows.

$$U={U_1}+{U_2}+{U_3}+{U_4}+{U_c}+{U_5}$$

3

It includes the elastic potential energy of five supports and the potential energy of coupling.

\({U_i}=\frac{1}{2}{\mathbf{q}}_{{{B_i}}}^{T}{\mathbf{K}}{{\mathbf{q}}_{{B_{_{i}}}}}=\frac{1}{2}{\left\{ {{{\mathbf{B}}_i}{{\mathbf{q}}_i}} \right\}^T}{\mathbf{K}}\left\{ {{{\mathbf{B}}_i}{{\mathbf{q}}_i}} \right\}\) , \(i=1\sim 4\) (4)

The elastic potential energy of the intermediate bearing is as follows.

$${U_5}=\frac{1}{2}{\left( {{{\mathbf{q}}_{{C_H}}} - {{\mathbf{q}}_{{C_L}}}} \right)^T}{{\mathbf{K}}_C}\left( {{{\mathbf{q}}_{{C_H}}} - {{\mathbf{q}}_{{C_L}}}} \right)$$

5

The elastic potential energy at the coupling is as follows.

$${U_C}=\frac{1}{2}{\left( {{{\mathbf{q}}_{{C_2}}} - {{\mathbf{q}}_{{C_1}}}} \right)^T}{{\mathbf{K}}_C}\left( {{{\mathbf{q}}_{{C_2}}} - {{\mathbf{q}}_{{C_1}}}} \right)=\frac{1}{2}{\left\{ {{{\mathbf{B}}_5}{{\mathbf{q}}_2} - {{\mathbf{B}}_4}{{\mathbf{q}}_1}} \right\}^T}{{\mathbf{K}}_C}\left\{ {{{\mathbf{B}}_5}{{\mathbf{q}}_2} - {{\mathbf{B}}_4}{{\mathbf{q}}_1}} \right\}$$

6

### The Excitation Mechanism Of The Support Misalignment

The characteristics of the slender shaft of the low-pressure rotor make the assembly more difficult, which increases the possibility of the support misalignment. The misalignment of the rear support of the low-pressure turbine rotor in the dual-rotor system will not only change the potential energy of the rotor system but also affect coupling which connecting the fan rotor and the low-pressure turbine rotor. The structure diagram of the low-pressure turbine rear support misalignment is shown in Fig. 2.

When the support has a misalignment of \(\Delta y\) in the *y*-direction and a misalignment of \(\Delta z\) in the *z-*direction, the support has a deformation in the equilibrium state, the expression of the elastic potential energy of the support becomes:

$${U_6}=\frac{1}{2}({\mathbf{q}}_{{{{\mathbf{B}}_{\mathbf{6}}}}}^{{\mathbf{T}}} - {\Delta ^T}){{\mathbf{K}}_{{B_6}}}({{\mathbf{q}}_{{{\mathbf{B}}_{\mathbf{6}}}}} - \Delta )$$

7

where \(\Delta ={(\begin{array}{*{20}{c}} 0&{\Delta y}&{\Delta z}&0&0 \end{array})^T}\). \({{\mathbf{K}}_{{{\mathbf{B}}_{\mathbf{6}}}}}\) is the stiffness matrix of misalignment support. \({{\mathbf{q}}_{{{\mathbf{B}}_{\mathbf{6}}}}}\) is the generalized displacement of the rear support of the low-pressure turbine disc.

When the low-pressure rotor is a misalignment, the sleeve coupling on the low-pressure rotor is bent, and there is a misalignment angle \(\alpha\) between the fan rotor and the low-pressure turbine rotor. The rotational angular velocities of the fan rotor and the low-pressure turbine rotor are \({\Omega _1}\)and \({\Omega _2}\)respectively. Fan rotor drives the low-pressure turbine rotor through the coupling when the angular displacement of the fan rotor is *ϕ**1*, the angle of rotation of the coupling gear sleeve is *ϕ**2*, and the relationship between *ϕ**1* and *ϕ**2* is as follows.

$$\operatorname{tg} {\phi _1}=\operatorname{tg} {\phi _2}\cos \alpha$$

8

Considering the small misalignment angle \(\alpha\), i.e \(\tan \alpha \cong \alpha\).

Derivatives on both sides of the above Eq. (8).

$$\frac{{{\Omega _2}}}{{{\Omega _1}}}=\frac{C}{{1+D\cos 2{\phi _1}}}$$

9

where \(C=\frac{{4\cos \alpha }}{{3+\cos 2\alpha }}\), \(D=\frac{{\cos 2\alpha - 1}}{{3+\cos 2\alpha }}\). The calculation method refer to Wang Meiling (2013). The angular acceleration of the low-pressure turbine rotor can be obtained by derivation of this formula:

$${\dot {\Omega }_2}=\frac{{2CD{\Omega _1}^{2}\sin 2{\phi _1}}}{{{{(1+Dcos2{\phi _1})}^2}}}$$

10

The torque transmitted from the fan rotor shaft to the low-pressure turbine rotor shaft through the coupling is mainly used for two parts, one part is the accelerated motion of the low-pressure turbine rotor, the other part is used to balance the resistance suffered by the low-pressure turbine rotor, namely:

where \({T_a}=I{\dot {\Omega }_2}\),\({T_f}=c {\Omega _2}\). is the moment of inertia of the low-pressure turbine rotor. *c* is the damping of the rotor, and the torque of the fan rotor is in the horizontal direction. When torque passes through the coupling, it divides into two parts:

\({T_x}=T\cos \alpha\) , \({T_s}=T\sin \alpha\) (12)

Combining Eq. (11) and Eq. (12), it can be concluded that:

$$T\cos \alpha =I{\dot {\Omega }_2}+c {\Omega _2}$$

13

where \({T_s}\) is in the axial direction of the low-pressure turbine rotor. Further, decompose \({T_s}\) into *y* and *z* directions.

\({T_y}=T\sin \alpha \sin \beta\) , \({T_z}=T\sin \alpha \cos \beta\) (14)

where \(\beta\) is the intersection angle between the misalignment vector and the *z*-axis. The torque forces of the low-pressure turbine rotor in the *y* and *z* directions at the coupling can be obtained.

### Misalignment Excitation Force

In the case of misalignment, the speed of the low-pressure turbine rotor has changed, and its excitation force also changes accordingly. The expression of the misalignment excitation force as follows.\({{\mathbf{Q}}_{2u}}=\left\{ {\begin{array}{*{20}{c}} { - \alpha {m_2}{e_2}{{\left[ {\frac{{C{\Omega _1}}}{{1+D\cos (2{\Omega _1}t)}}} \right]}^2}\cos (\frac{{C{\Omega _1}t}}{{1+D\cos (2{\Omega _1}t)}}+{\varphi _{20}})} \\ {{m_2}{e_2}{{\left[ {\frac{{C{\Omega _1}}}{{1+D\cos (2{\Omega _1}t)}}} \right]}^2}\cos (\frac{{C{\Omega _1}t}}{{1+D\cos (2{\Omega _1}t)}}+{\varphi _{20}})} \\ {{m_2}{e_2}{{\left[ {\frac{{C{\Omega _1}}}{{1+D\cos (2{\Omega _1}t)}}} \right]}^2}\sin (\frac{{C{\Omega _1}t}}{{1+D\cos (2{\Omega _1}t)}}+{\varphi _{20}})} \\ 0 \\ 0 \end{array}} \right\}\) (15)

From the Eq. 15, it can be concluded that the excitation frequency of the low-pressure turbine disc is \(\frac{C}{{1+D\cos (2{\Omega _1}t)}}\) times of rotating frequency of the fan rotor. And the amplitude of excitation force is directly proportional to the sum of unbalance \({m_2}{e_2}\) of the low-pressure turbine disc and \({\left[ {\frac{{C{\Omega _1}}}{{1+D\cos (2{\Omega _1}t)}}} \right]^2}\).

### Vibration Of The Dual-rotor Excited By Both Unbalance And Support Misalignment

## Misalignment in Z-direction

The misalignment of the rear support of the low-pressure turbine included the *y-*direction misalignment, *z*-direction misalignment, and comprehensive misalignment in the two directions. Four measuring points are selected on the low-pressure and the high-pressure rotor respectively, including 1# support (No.1), the centroid of LPC (No.2), the centroid of LPT (No.3), 6# support (No.4), 4# support (No.5), the centroid of HPC (No.6), the centroid of HPT (No.7) and the intermediate bearing (No.8). The rotational speed of the low-pressure rotor and the high-pressure rotor are 5000r/min (N1) and 6000r/min (N2) respectively, and the initial unbalance of the fan disc is 3.0×10− 3kg·m.

The situation that misalignment of 0.5mm exists in the *z*-direction at 6# supporting was analyzed, and the time-domain responses, the frequency spectra, and the shaft-center trajectories of each measuring point for the dual-rotor system are obtained, as showed in Fig. 3.

It can be seen from the frequency spectra that the vibration in the y-direction at 1#, 2#, 3#, 4#, 7# and 8# measuring point has two obvious frequencies, one is rotational frequency of the low-pressure rotor (N1), the other one is 2 times of rotational frequency of the low-pressure rotor (2N1). The orbit of the axis at the center is not a regular circle, but approximate ‘concave’ shape, which indicates that the vibration response in the y-direction and the z-direction is coupled. The measuring points 5# and 6# on the high-pressure rotor are less affected by the misalignment, and the amplitude of the 2 times of rotational frequency of the low-pressure rotor (2N1) in the spectrum diagram is very small, which can be ignored, and the orbit of the axis center is a regular circle.

### Y-direction And Z-direction Misalignment

The situation that the misalignment of 0.5mm exists in the z-direction and y-direction at 6# supporting was analyzed, and the time-domain responses, the frequency spectra, and the shaft-center trajectories of each measuring point for the dual-rotor system are obtained, as shown in Fig. 4.

According to the vibration response of 8 measuring points, the following conclusions can be drawn:

(1) It can be seen from the spectrum diagram that the misalignment of the rear support of the low-pressure turbine will cause the 2 times rotational frequency of the low-pressure rotor.

(2) It can be seen from the axle center trajectory in the figure that ‘concave’ axle center trajectory appears at measuring point 1 and point 2; The axial trajectories of measuring point 3, point 4, point 7 and point 8 are an external ‘8’ shape. And axial trajectories of measuring points 5 and 6 are close to circular. This is because measuring points 3, 4, 7 and 8 are close to the rear support of the misalignment low-pressure turbine while measuring points 5 and 6 are coupled with the low-pressure rotor to generate frequency 2N1 and are far away from the rear support of the misalignment low-pressure turbine, so they are least affected by misalignment.

(3) It can be seen from the frequency spectra that the magnitude of frequency N1 of 5# and 6# is larger, which indicates that the two measuring points are greatly affected by unbalanced excitation.

### Influence Of Misalignment On The Vibration Response Of The Dual-rotor System

In this section, the variation law of the dual-rotor system vibration with misalignment of the support is studied and the fitting curves of vibration amplitude of each measuring point of the dual-rotor system vary with misalignment are obtained, as showed in Fig. 5 and Table 5. Among them, Fig. 5(a) is a three-dimensional spectrum diagram of the dual-rotor system vibration with different misalignment.

Table 5

Fitting curve equation of vibration amplitude with misalignment

Misalignment (mm) | 2N1(µm) | Fitting function value(µm) y = 65.5x2-0.041x-0.01 | Error |

0 | 0 | -0.01 | - |

0.2 | 2.59 | 2.60 | 0.45% |

0.4 | 10.45 | 10.45 | 0.03% |

0.6 | 23.54 | 23.55 | 0.02% |

0.8 | 41.88 | 41.88 | 0.01% |

1.0 | 65.45 | 65.45 | 0.01% |

1.2 | 94.26 | 94.26 | 0.01% |

1.4 | 128.3 | 128.3 | 0.01% |

1.6 | 167.6 | 167.6 | 0.01% |

1.8 | 212.1 | 212.1 | 0.02% |

2.0 | 261.9 | 261.9 | 0.01% |

Figure 5(b) is a fitting curve of the vibration amplitude of frequency 2N1 varying with the size of misalignment. The fitting curve equation of the vibration amplitude of frequency 2N1 changing with misalignment of the dual-rotor system is shown in Table 5.

It can be seen from Fig. 5(a) that with the increase of the misalignment of the support, the amplitude of frequency N1 has no obvious change, while the amplitude of frequency 2N1 increases. From Fig. 5(b), it can be seen that the vibration amplitude of frequency 2N1 is approximately quadratic with the misalignment. And the fitting curve equation and fitting error are obtained by fitting the quadratic curve, as showed in Table 5. Through analysis, it is found that the fitting error is very small. It can be considered that the amplitude of the rotor vibration frequency 2N1 caused by the misalignment of the rear support of the low-pressure turbine has a quadratic relationship with the misalignment.

### Influence Of Rotor Load Torque On The Vibration Response Of The Dual-rotor System

The influence of load torque on the response of the dual-rotor support misalignment is studied in this section. The three-dimensional spectrum of vibration amplitude for every test point varying with load torque are obtained, as showed in Fig. 6(a). The relationship between the vibration response amplitude of the dual-rotor system and the load torque of the rotor is analyzed, and the fitting curve is obtained, as showed in Fig. 6(b). And the fitting equation and error are shown in Table 6.

Table 6

Fitting curve equation of vibration amplitude with load torque

load torque(N*m) | 2N1(µm) | Fitting function value(µm) y = 32.84x-0.03 | Error |

0.5 | 16.41 | 16.39 | 0.061% |

1.0 | 32.80 | 32.81 | 0.030% |

1.5 | 49.24 | 49.23 | 0.020% |

2.0 | 65.66 | 65.65 | 0.015% |

2.5 | 82.08 | 82.07 | 0.012% |

3.0 | 98.51 | 98.49 | 0.010% |

3.5 | 114.90 | 114.91 | 0.008% |

4.0 | 131.30 | 131.33 | 0.023% |

4.5 | 147.81 | 147.75 | 0.034% |

As can be seen from Fig. 6(a), the amplitude of vibration frequency N1 of the rotor system does not change with the increase of rotor load torque. And the amplitude of the vibration frequency 2N1 is approximately linearly related to the load torque as showed in Fig. 6(b). The linear fitting results between the amplitude of frequency 2N1 and the load torque are shown in Table 6. It can be seen that the error of the fitting equation is within 0.1%. It can be considered that the amplitude of the rotor vibration frequency 2N1 caused by the misalignment of the rear support of the low-pressure turbine is linearly related to the load torque.

### Influence Of Unbalance On Vibration Response Of The Dual-rotor System

In this section, the influence of unbalanced mass on the vibration response of the dual-rotor system with support misalignment faults is studied. The three-dimensional spectrum of the dual-rotor vibration response varying with the unbalance of the fan disc is plotted, as showed in Fig. 7(a). The fitting curve of the amplitude of frequency N1 with unbalance in the vibration response of the dual-rotor system is obtained, as showed in Fig. 7(b). And the fitting curve equation and fitting error between the amplitude of the vibration frequency N1 and the magnitude of the unbalance are obtained by fitting the quadratic curve, as showed in Table 7.

Table 7

Fitting curve equation of vibration amplitude with unbalance

x | N1(µm) | Fitting function(µm) y = 12.92x-0.18 | Error |

0.6 | 7.73 | 7.57 | 2.04% |

1.2 | 15.49 | 15.32 | 1.07% |

1.8 | 23.24 | 23.08 | 0.71% |

2.4 | 30.99 | 30.83 | 0.52% |

3.0 | 38.74 | 38.58 | 0.41% |

3.6 | 46.50 | 46.33 | 0.36% |

4.2 | 54.25 | 54.08 | 0.31% |

4.8 | 62.02 | 61.84 | 0.26% |

5.4 | 69.75 | 69.59 | 0.23% |

As can be seen from Fig. 7(a), the amplitude of the vibration response frequency N1 of the dual-rotor system gradually increases with the increase of the unbalanced mass, the amplitude of 2N1 does not change. Furthermore, the fitting curve of the vibration amplitude of the frequency N1 with the unbalanced mass is given in Fig. 7(b), it is found that the amplitude of the frequency N1 has a linear relationship with the magnitude of the unbalanced mass. From the fitting results in Table 7, it can be seen that the error of the fitting function gradually decreases with the increase of unbalanced mass. It can be considered that the amplitude of the frequency N1 is linearly related to the unbalanced mass.